Using the Corresponding Angles Theorem, Vertical Angles Theorem, and the Transitive Property of Congruence, we can prove that alternate exterior angles (e.g, <4 and <5) are congruent by the alternate exterior angles theorem.
Recall:
- Alternate exterior angles are angles that lie outside the two lines that is cut across by a transversal but on opposite sides along the transversal.
- Examples of alternate exterior angles are <2 and <7; <4 and <5 as shown in the figure attached below.
If we are given that [tex]m \parallel n[/tex] in the diagram attached below, the following are theorems and definitions we can use to prove that [tex]\angle 4 \cong \angle 5[/tex] (alternate exterior angles).
Statement 1: [tex]\angle 4 \cong \angle 8[/tex]
Reason: Corresponding Angles Theorem
The corresponding angles theorem states that when two parallel lines (lines m and n) are intersected by a transversal line (line w), the two corresponding angles formed (e.g. <4 and <8) are congruent.
Statement 2: [tex]\angle 8 \cong \angle 5[/tex]
Reason: Vertical Angles Theorem
The Vertical Angles Theorem states that the opposite vertical angles (e.g. <8 and <5) formed when two lines (lines n and w) intersect are congruent to each other.
Statement 3: [tex]\angle 4 \cong \angle 5[/tex]
Reason: Transitive Property of Congruence
The Transitive Property of Congruence states that if a = b; and b = c; then a = c.
Therefore, using the Corresponding Angles Theorem, Vertical Angles Theorem, and the Transitive Property of Congruence, we can prove that alternate exterior angles (e.g, <4 and <5) are congruent by the alternate exterior angles theorem.
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